Chapter 8

Write the first five terms of each arithmetic sequence. Do not use a calculator. $$a_{1}=5, d=-2$$

Evaluate the following. In Exercises 17 and \(18,\) express the answer in terms of \(n .\) Do not use a calculator. $$\left(\begin{array}{c}13 \\\13\end{array}\right)$$

Find \(a_{5}\) and \(a_{n}\) for each geometric sequence. $$a_{4}=243, r=-3$$

Use mathematical induction to prove each statement. Assume that \(n\) is a positive integer. $$3+3^{2}+3^{3}+\cdots+3^{n}=\frac{3\left(3^{n}-1\right)}{2}$$

Write the first five terms of each sequence. Do not use a calculator. $$a_{n}=\frac{4 n-1}{n^{2}+2}$$

Write the first five terms of each arithmetic sequence. Do not use a calculator. $$a_{1}=4, d=3$$

Find \(a_{5}\) and \(a_{n}\) for each geometric sequence. $$a_{4}=18, r=2$$

Use mathematical induction to prove each statement. Assume that \(n\) is a positive integer. $$1^{2}+2^{2}+3^{2}+\cdots+n^{2}=\frac{n(n+1)(2 n+1)}{6}$$

Write the first five terms of each sequence. Do not use a calculator. $$a_{n}=\frac{n^{2}-1}{n^{2}+1}$$

Evaluate the following. In Exercises 17 and \(18,\) express the answer in terms of \(n .\) Do not use a calculator. $$\left(\begin{array}{c}12 \\\12\end{array}\right)$$

Evaluate each expression. $$4 ! \cdot 5$$

Write the first five terms of each arithmetic sequence. Do not use a calculator. $$a_{3}=10, d=-2$$

Find \(a_{5}\) and \(a_{n}\) for each geometric sequence. $$-4,-12,-36,-108, \dots$$

A student gives the answer to a probability problem as \(\frac{6}{5}\). Explain why this answer must be incorrect.

Use mathematical induction to prove each statement. Assume that \(n\) is a positive integer. $$1^{3}+2^{3}+3^{3}+\cdots+n^{3}=\frac{n^{2}(n+1)^{2}}{4}$$

Your friend does not understand what is meant by the \(n\) th term, or general term, of a sequence. How would you explain this idea?

Evaluate the following. In Exercises 17 and \(18,\) express the answer in terms of \(n .\) Do not use a calculator. $$\left(\begin{array}{l}8 \\\3\end{array}\right)$$

Write the first five terms of each arithmetic sequence. Do not use a calculator. $$a_{1}=3-\sqrt{2}, a_{2}=3$$

Find \(a_{5}\) and \(a_{n}\) for each geometric sequence. $$-2,6,-18,54, \dots$$

If the probability of an event is 0.857 what is the probability that the event will not occur?

Use mathematical induction to prove each statement. Assume that \(n\) is a positive integer. $$5 \cdot 6+5 \cdot 6^{2}+5 \cdot 6^{3}+\cdots+5 \cdot 6^{n}=6\left(6^{n}-1\right)$$

If \(n\) is a positive integer greater than \(1,\) is \((n-1) ! \cdot n\) always equal to \(n ! ?\)

Evaluate the following. In Exercises 17 and \(18,\) express the answer in terms of \(n .\) Do not use a calculator. $$\left(\begin{array}{l}9 \\\7\end{array}\right)$$

Find \(a_{8}\) and \(a_{n}\) for each arithmetic sequence. $$a_{1}=5, d=2$$

Evaluate the following. In Exercises 17 and \(18,\) express the answer in terms of \(n .\) Do not use a calculator. $$\left(\begin{array}{c}100 \\\2\end{array}\right)$$

Find \(a_{5}\) and \(a_{n}\) for each geometric sequence. $$\frac{4}{5}, 2,5, \frac{25}{2}, \dots$$

Work each problem. A marble is drawn at random from a box containing 3 yellow, 4 white, and 8 blue marbles. Find the probabilities in parts (a)-(c). (a) A yellow marble is drawn. (b) A black marble is drawn. (c) The marble is yellow or white. (d) What are the odds in favor of drawing a yellow marble? (e) What are the odds against drawing a blue marble?

Use mathematical induction to prove each statement. Assume that \(n\) is a positive integer. $$\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\dots+\frac{1}{n(n+1)}=\frac{n}{n+1}$$

Evaluate each expression. $$P(7,7)$$

Decide whether each sequence is finite or infinite. The sequence of days of the week

Find \(a_{8}\) and \(a_{n}\) for each arithmetic sequence. $$a_{1}=-3, d=-4$$

Find \(a_{5}\) and \(a_{n}\) for each geometric sequence. $$\frac{1}{2}, \frac{2}{3}, \frac{8}{9}, \frac{32}{27}, \ldots$$

A baseball player with a batting average of .300 comes to bat. What are the odds in favor of his getting a hit?

Use mathematical induction to prove each statement. Assume that n is a positive integer. $$7 \cdot 8+7 \cdot 8^{2}+7 \cdot 8^{3}+\dots+7 \cdot 8^{n}=8\left(8^{n}-1\right)$$

Evaluate each expression. $$P(5,5)$$

Decide whether each sequence is finite or infinite. The sequence of dates in the month of November

Evaluate the following. In Exercises 17 and \(18,\) express the answer in terms of \(n .\) Do not use a calculator. $$\left(\begin{array}{c}20 \\\15\end{array}\right)$$

Find \(a_{8}\) and \(a_{n}\) for each arithmetic sequence. $$a_{3}=2, d=1$$

Find \(a_{5}\) and \(a_{n}\) for each geometric sequence. $$10,-5, \frac{5}{2},-\frac{5}{4}, \dots$$

Use mathematical induction to prove each statement. Assume that n is a positive integer. $$\frac{4}{5}+\frac{4}{5^{2}}+\frac{4}{5^{3}}+\dots+\frac{4}{5^{n}}=1-\frac{1}{5^{n}}$$

Evaluate each expression. $$P(9,2)$$

Evaluate the following. In Exercises 17 and \(18,\) express the answer in terms of \(n .\) Do not use a calculator. $$\left(\begin{array}{l}5 \\\0\end{array}\right)$$

Decide whether each sequence is finite or infinite. $$1,2,3,4$$

Find \(a_{8}\) and \(a_{n}\) for each arithmetic sequence. $$a_{4}=5, d=-2$$

Find \(a_{5}\) and \(a_{n}\) for each geometric sequence. $$3,-\frac{9}{4}, \frac{27}{16},-\frac{81}{64}, \dots$$

If the odds that it will rain are 4 to5, what is the probability of rain?

Use mathematical induction to prove each statement. Assume that n is a positive integer. $$\frac{1}{2}+\frac{1}{2^{2}}+\frac{1}{2^{3}}+\dots+\frac{1}{2^{n}}=1-\frac{1}{2^{n}}$$

Evaluate each expression. $$P(10,3)$$

Evaluate the following. In Exercises 17 and \(18,\) express the answer in terms of \(n .\) Do not use a calculator. $$\left(\begin{array}{l}6 \\\0\end{array}\right)$$

Decide whether each sequence is finite or infinite. $$-1,-2,-3,-4$$